The present invention is in the field of dynamic balancing techniques and specifically relates to a procedure for balancing relatively long shafts for use at very high rotational speeds.
Typical of the shafts to which the method of the present invention applies are the driveshafts used in helicopters to transmit hundreds of horsepower from the gearbox adjacent the engine to the tail rotor. Typically, such a shaft is an aluminum tube of 4.5 inches outside diameter, 16 feet long, having a wall thickness of 0.065 inches, and transmitting several hundred horsepower at 8000 rpm.
There is a limit to how much power a shaft of a given mass can transmit safely. Normally there is a trade-off between a relatively short and thick shaft transmitting a large torque at low speed or a relatively thin and long shaft transmitting a small torque and rotating at high speed. The present invention is concerned with the latter alternative because the distance is relatively large. Therefore, the shaft is rotating at a relatively high speed. To save weight, the driveshaft is in the form of a hollow tube.
An ideal driveshaft would be assembled from a tube that is absolutely round, absolutely straight, and has uniformly thick walls. Unfortunately, this condition is never found in actual practice. In practice, the cross sections of the tubes are distorted, the tubes are bowed, and the walls include thicker and thinner spots.
When a shaft is long relative to its diameter, it is more susceptible to whipping caused by an unbalanced mass distribution, and this is aggravated as the speed increases. Although some of the defects in the tube are of a relatively minor nature, at high speed the effect of the relatively minor imperfections is quite pronounced, indicating that careful balancing of the shaft is necessary to run at such high speeds.
The natural frequencies of a simply supported shaft are given by the following equation: ##EQU1## where; .omega..sub.n =n.sup.th natural frequency (rad -sec.sup.-1)
n=Mode number (1, 2, 3, etc.) PA1 .mu..sub.1 =Mass per unit length of shaft (lb in. .sup.-2 sec.sup.2) PA1 E=modulus of elasticity (lb in. .sup.-2) PA1 I=moment of inertia of section of shaft about a diameter (in..sup.4) PA1 l=Length of shaft (in.)
The rotational speed of the shaft equal to the fundamental frequency of vibration of the shaft is also called the critical speed of the shaft corresponding to operation in mode 1, i.e., the fundamental frequency of the shaft. The present invention is concerned with shafts that operate at speeds between the second and third critical speeds (modes no. 2 and 3). As the shaft spins up to speed it must pass through the first and second critical speeds, corresponding to modes no. 1 and 2. Unbalance of the shaft is particularly detrimental when the shaft is rotating at any of the critical speeds, because vibrations resulting from shaft unbalance are magnified at those speeds.
As designers become more weight and vibration conscious for new aircraft designs, the introduction of power-transmitting shafts that run at supercritical speeds becomes increasingly attractive. Balancing such high-speed shafts so they run smoothly has long been a problem, with the process being both difficult and costly. It traditionally requires many trial runs over the speed spectrum to achieve good balance.